BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures

Abstract

Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray--Singer holomorphic analytic torsions. To this end, extending work of Fang--Lu--Yoshikawa in dimension 3, we introduce and study the BCOV invariant of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of its behaviour at the boundary of moduli spaces is imperative. We address this problem by proving precise asymptotics along one-parameter degenerations, in terms of topological data and intersection theory. Central to the approach are new results on degenerations of $L^2$ metrics on Hodge bundles, combined with information on the singularities of Quillen metrics in our previous work.

arXiv commentMinor revision. Mainly restructure of the text, minor improvements and corrections. Added information about subdominant terms of $L^2$-norms

References in corpus (5)

• Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes
• Enumerative geometry of Calabi-Yau 4-folds
• The Hitchin functionals and the topological B-model at one loop
• Combinatorial degenerations of surfaces and Calabi--Yau threefolds
• Singularities of metrics on Hodge bundles and their topological invariants

Cited by in corpus (4)

• On genus one mirror symmetry in higher dimensions and the BCOV conjectures
• Motivic integration and the birational invariance of BCOV invariants
• Boundary asymptotics of the relative Bergman kernel metric for curves
• Calabi-Yau/Landau-Ginzburg Correspondence for Weil-Peterson Metrics and $t{t}^{\ast }$ Structures